The Tits isomorphism is noncanonical

Assume the Axiom of Choice, used through Tits deformation (Tits deformation for the type-A Hecke algebra, The Axiom of Choice). The isomorphism $H=e_B\mathbb{C}[G]e_B\cong \mathbb{C}[S_n]$ supplied by Tits deformation, and the consequent parametrisation of the constituents of a principal series by tuples of partitions, are not canonical: the deformation argument produces an isomorphism by deforming through the generic algebra, using an open subset of the parameter line, lifting idempotents, determinant inversion and a constructible incidence locus, and it does not canonically identify the standard basis $(T_w)$ with the group elements $(w)$ nor the simple $H$-modules with the Specht modules (The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n). Indeed there is generally no algebra isomorphism sending every standard $T_w$ to $w$ when $q\mathrm{\ne }1$, since the quadratic relations $T_{s_i}^2=(q-1)T_{s_i}+q1$ and $s_i^2=1$ differ (The finite Hecke algebra is non-canonically isomorphic to the group algebra of S_n). Different choices in the deformation (or different specialisations of the generic algebra) can produce different isomorphisms, and downstream arguments must not treat the partition labelling of a single constituent as if it were canonical or compatible with every natural operation. What is canonical is the resulting numerical data: the number of simple constituents and the multiset of their multiplicities as dimensions of simple $\mathbb{C}[W_{\chi }]$-modules, as recorded in The constituents of a general finite principal series.